 9.8.1: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.2: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.3: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.4: Use the Liapunov function V(x, y, z) = x2 + y2 + z2 to show that th...
 9.8.5: Consider the ellipsoid V(x, y, z) = rx2 + y2 + (z 2r) 2 = c > 0. (a...
 9.8.6: In each of 6 through 10 carry out the indicated investigations of t...
 9.8.7: In each of 6 through 10 carry out the indicated investigations of t...
 9.8.8: In each of 6 through 10 carry out the indicated investigations of t...
 9.8.9: In each of 6 through 10 carry out the indicated investigations of t...
 9.8.10: In each of 6 through 10 carry out the indicated investigations of t...
 9.8.11: (a) Show that there are no critical points when c < 0.5, one critic...
 9.8.12: (a) Let c = 1.3. Find the critical points and the corresponding eig...
 9.8.13: The limit cycle found in comes into existence as a result of a Hopf...
 9.8.14: (a) Let c = 3. Find the critical points and the corresponding eigen...
 9.8.15: (a) Let c = 3.8. Find the critical points and the corresponding eig...
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations includes 15 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations have been answered, more than 13835 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.